A235189 Number of ways to write n = (1 + (n mod 2))*p + q with p < n/2 such that p, q and prime(p) - p + 1 are all prime.

0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 4, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 4, 2, 2, 4, 4, 1, 3, 2, 3, 2, 3, 3, 4, 3, 4, 3, 3, 3, 5, 2, 4, 4, 2, 2, 6, 2, 2, 4, 1, 1, 5, 4, 5, 4, 4, 2, 4, 3, 3, 3, 4, 4, 5, 4, 5, 4, 3, 2, 4, 2, 3, 6, 5, 3, 6, 3, 5, 5, 2, 3, 9, 3, 3, 5, 3, 1, 6, 3

Offset

1,9

Comments

Conjecture: a(n) > 0 for all n > 6.

This implies both Goldbach's conjecture (A045917) and Lemoine's conjecture (A046927). For primes p with prime(p) - p + 1 also prime, see A234695.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Example

a(10) = 1 since 10 = 3 + 7 with 3, 7 and prime(3) - 3 + 1 = 3 all prime.
a(28) = 1 since 28 = 5 + 23 with 5, 23 and prime(5) - 4 = 7 all prime.
a(61) = 1 since 61 = 2*7 + 47 with 7, 47 and prime(7) - 6 = 11 all prime.
a(98) = 1 since 98 = 31 + 67 with 31, 67 and prime(31) - 30 = 97 all prime.

Mathematica

p[n_]:=PrimeQ[Prime[n]-n+1]
a[n_]:=Sum[If[p[Prime[k]]&&PrimeQ[n-(1+Mod[n, 2])*Prime[k]], 1, 0], {k, 1, PrimePi[(n-1)/2]}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000040, A045917, A046927, A234694, A234695, A235187.

Sequence in context: A161056 A161260 A161285 * A308452 A103682 A023134

Adjacent sequences: A235186 A235187 A235188 * A235190 A235191 A235192

Keyword

nonn

Author

Zhi-Wei Sun, Jan 04 2014

Status

approved